Creation

Alliance graph is a set of chapters-nodes which created based on “ally” and “opponent” connections. It is created using the following data from taoteprog (which I manually created while reading a book):

• ttp_chapters - a tibble with chapter data (column chapter for its number and name for its name). The whole list of chapters can be found here.
• ttp_edges - a tibble with data about chapter connections. It has the following columns:
  • chapter1 for the chapter number at which list of connections is printed.
  • chapter2 for the number of connected chapter.
  • type for connection type (either “ally” or “opponent”).
  Note that connections between chapters are not symmetrical, i.e. one chapter can be an “ally” or “opponent” of the another but not the other way around.

ttp_raw <- tbl_graph(nodes = ttp_chapters, edges = ttp_edges, directed = TRUE)
ttp_raw
## # A tbl_graph: 81 nodes and 124 edges
## #
## # A directed multigraph with 20 components
## #
## # Node Data: 81 x 2 (active)
##   chapter name                   
##     <int> <chr>                  
## 1       1 Program                
## 2       2 Program Well           
## 3       3 Think Chess            
## 4       4 Carve Reality          
## 5       5 Solve the Problem      
## 6       6 Don't Solve the Problem
## # ... with 75 more rows
## #
## # Edge Data: 124 x 3
##    from    to type    
##   <int> <int> <chr>   
## 1     4     9 ally    
## 2     4    54 ally    
## 3     5     6 opponent
## # ... with 121 more rows

The main data structure of tidygraph is a tbl_graph which can be thought of as a combination of tibbles with data for nodes and edges. Note that values in from and to columns in edges tibble are the row indices of the nodes stored in nodes tibble, which will be a little confusing when nodes won’t cover all chapters.

Important features of this graph:

• There are 2 loops in graph (chapters “Be Consistent”, “Follow The Way”). And all of them are of type “opponent”, which I consider to be a subtle joke rather than editor mistake. However, for exploration these edges will be removed.
• There are 20 [weak] components in this graph (maximal groups of nodes where each pair can be connected by undirected path). The number is this big because there are many chapters with no “allies” and no “opponents” (isolated points in graph), which will be also removed for exploration.

So the graph of interest is constructed as follows:

ttp <- ttp_raw %>%
  # Remove loops
  activate(edges) %>%
  filter(!edge_is_loop()) %>%
  # Remove isolated nodes
  activate(nodes) %>%
  filter(!node_is_isolated())
ttp
## # A tbl_graph: 67 nodes and 122 edges
## #
## # A directed simple graph with 6 components
## #
## # Node Data: 67 x 2 (active)
##   chapter name                   
##     <int> <chr>                  
## 1       4 Carve Reality          
## 2       5 Solve the Problem      
## 3       6 Don't Solve the Problem
## 4       7 Enjoy Confusion        
## 5       8 Procrastinate          
## 6       9 Verbalize and Nounalize
## # ... with 61 more rows
## #
## # Edge Data: 122 x 3
##    from    to type    
##   <int> <int> <chr>   
## 1     1     6 ally    
## 2     1    44 ally    
## 3     2     3 opponent
## # ... with 119 more rows

Some explanations of the magic just happened:

• activate() verb from tidygraph serves as a switch between tibbles for nodes and edges. All dplyr verbs applied to tbl_graph object are applied to the active tibble.
• edge_is_loop(), node_is_isolated() and group_components() are functions of the same essence as n() in dplyr: they both should be called inside a graph computation functions.

Visualization

ttp %>%
  ggraph(layout = "nicely") +
    geom_edge_link(
      aes(colour = type),
      arrow = arrow(length = unit(1.5, "mm")),
      start_cap = circle(3, "mm"),
      end_cap = circle(3, "mm")
    ) +
    geom_node_text(aes(label = chapter), size = 5) +
    scale_edge_colour_manual(values = c(ally = "#22B022",
                                        opponent = "#A4AAF6")) +
    theme_graph() +
    labs(
      title = '"Tao Te Programming" alliance graph',
      subtitle = "Nodes represent chapter numbers, edges - connections",
      caption = "@echasnovski"
    )

Important notes about creation of this plot:

• ggraph() is the equivalent of ggplot() in ggplot2 and also returns a ggplot object. ggraph() takes a layout argument for the type of graph arrangement in plot.
• geom_edge_link() and geom_node_text() are very much ggplot2-like functions: they create layers of plot (for links and nodes respectively).
• theme_graph() is a pre-installed ggraph theme.

Exploration

Notes about graph structure:

⬛ There are two relatively big components (with 16 and 40 nodes) and 4 small ones (one with 5 nodes and three with 2).

⬛ There are considerably more “ally” connections then “opponent” which can be confirmed by the following code:

ttp_edges %>% count(type)
## # A tibble: 2 x 2
##   type         n
##   <chr>    <int>
## 1 ally        88
## 2 opponent    36

⬛ The graph itself can be considered as signed graph, i.e. a graph “in which each edge has a positive or negative sign”. This type of graphs can be studied for special community detection: nodes within common community should be primarily connected with positive (“ally”) edges and between communities - by negative (“opponent”) ones.

Interesting problem description and solution are presented in this paper. I implemented the variation of the suggested approach, which can be found in my raw analysis on github (or here for its html output). Implementation is rather verbose with no very special insight, so I decided to not include it here.